Advertisement

Theoretical and Mathematical Physics

, Volume 197, Issue 3, pp 1727–1736 | Cite as

The Topology of Isoenergetic Surfaces for the Borisov–Mamaev–Sokolov Integrable Case on the Lie Algebra so(3, 1)

  • R. Akbarzadeh
Article
  • 4 Downloads

Abstract

We describe the topology of isoenergetic surfaces for an integrable system on the Lie algebra so(3, 1) and the critical points of the Hamiltonian for different parameter values. We construct bifurcation values of the Hamiltonian.

Keywords

topology integrable Hamiltonian system isoenergetic surface critical set bifurcation diagram 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A. T. Fomenko, “The theory of invariants of multidimensional integrable Hamiltonian systems (with arbitrary many degrees of freedom): Molecular table of all integrable systems with two degrees of freedom,” in: Topological Classification of Integrable Systems (Adv. Sov. Math., Vol. 6, A. T. Fomenko, ed.), Amer. Math. Soc., Providence, R. I. (1991), pp. 1–35.CrossRefGoogle Scholar
  2. 2.
    A. T. Fomenko, “Morse theory of integrable Hamiltonian systems,” Sov. Math. Dokl., 33, 502–506 (1986).zbMATHGoogle Scholar
  3. 3.
    A. T. Fomenko, “Topological invariants of Liouville integrable Hamiltonian systems,” Funct. Anal. Appl., 22, 286–296 (1988).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    A. V. Bolsinov and A. T. Fomenko, Integrable Hamiltonian Systems: Geometry, Topology, Classification [in Russian], Vol. 1, Udmurtskii Univ. Press, Izhevsk (1999).zbMATHGoogle Scholar
  5. 5.
    A. A. Oshemkov, “Fomenko invariants for the main integrable cases of the rigid body motion equations,” in: Topological Classification of Integrable Systems (Adv. Sov. Math., Vol. 6, A. T. Fomenko, ed.), Amer. Math. Soc., Providence, R. I. (1991), pp. 67–146.Google Scholar
  6. 6.
    V. V. Sokolov, “One class of quadratic so(4) Hamiltonians,” Dokl. Math., 69, 108–111 (2004).zbMATHGoogle Scholar
  7. 7.
    V. V. Sokolov, “A new integrable case for the Kirchhoff equation,” Theor. Math. Phys., 129, 1335–1340 (2001).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    A. V. Borisov, I. S. Mamaev, and V. V. Sokolov, “A new integrable case on so(4),” Dokl. Phys., 46, 888–889 (2001).ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    A. V. Borisov and I. S. Mamaev, The Dynamics of a Rigid Body [in Russian], RKhD, Izhevsk (2001).Google Scholar
  10. 10.
    D. V. Novikov, “Topological features of the Sokolov integrable case on the Lie algebra e(3),” Sb. Math., 202, 749–781 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    D. V. Novikov, “The topology of isoenergy surfaces for the Sokolov integrable case on the Lie algebra so(3, 1),” Moscow Univ. Math. Bull., 66, 181–184 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    R. Akbarzadeh and G. Haghighatdoost, “The topology of Liouville foliation for the Borisov–Mamaev–Sokolov integrable case on the Lie algebra so(4),” Regul. Chaotic Dyn., 20, 317–344 (2015).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    R. Akbarzadeh, “Topological analysis corresponding to the Borisov–Mamaev–Sokolov integrable system on the Lie algebra so(4),” Regul. Chaotic Dyn., 21, 1–17 (2016).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    S. Smale, “Topology and mechanics I,” Invent. Math., 10, 305–331 (1970)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 14a.
    “Topology and mechanics II,” Invent. Math., 11, 45–64 (1970).Google Scholar
  16. 15.
    Ya. V. Tatarinov, “Equations of classical mechanics in new form,” Mosc. Univ. Mech. Bull., 58, No. 3, 13–22 (2003).Google Scholar
  17. 16.
    P. E. Ryabov, “Bifurcations of first integrals in the Sokolov case,” Theor. Math. Phys., 134, 181–197 (2003).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.School of MathematicsInstitute for Research in Fundamental Sciences (IPM)TehranIran

Personalised recommendations